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Theorem ontric3g 40146
 Description: For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥 ∈ 𝑦, 𝑦 = 𝑥, or 𝑦 ∈ 𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.)
Assertion
Ref Expression
ontric3g 𝑥 ∈ On ∀𝑦 ∈ On ((𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥𝑦𝑦𝑥)) ∧ (𝑦𝑥 ↔ ¬ (𝑥𝑦𝑦 = 𝑥)))
Distinct variable group:   𝑥,𝑦

Proof of Theorem ontric3g
StepHypRef Expression
1 orcom 867 . . . . . . 7 ((𝑦 = 𝑥𝑦𝑥) ↔ (𝑦𝑥𝑦 = 𝑥))
21a1i 11 . . . . . 6 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥𝑦𝑥) ↔ (𝑦𝑥𝑦 = 𝑥)))
3 onsseleq 6219 . . . . . 6 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
4 ontri1 6212 . . . . . 6 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
52, 3, 43bitr2d 310 . . . . 5 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥𝑦𝑥) ↔ ¬ 𝑥𝑦))
65con2bid 358 . . . 4 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)))
76ancoms 462 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)))
84ancoms 462 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
9 ontri1 6212 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
108, 9anbi12d 633 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑦𝑥𝑥𝑦) ↔ (¬ 𝑥𝑦 ∧ ¬ 𝑦𝑥)))
11 eqss 3968 . . . 4 (𝑦 = 𝑥 ↔ (𝑦𝑥𝑥𝑦))
12 ioran 981 . . . 4 (¬ (𝑥𝑦𝑦𝑥) ↔ (¬ 𝑥𝑦 ∧ ¬ 𝑦𝑥))
1310, 11, 123bitr4g 317 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 = 𝑥 ↔ ¬ (𝑥𝑦𝑦𝑥)))
14 equcom 2026 . . . . . . 7 (𝑦 = 𝑥𝑥 = 𝑦)
1514orbi2i 910 . . . . . 6 ((𝑥𝑦𝑦 = 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦))
1615a1i 11 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦𝑦 = 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦)))
17 onsseleq 6219 . . . . 5 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ (𝑥𝑦𝑥 = 𝑦)))
1816, 17, 93bitr2d 310 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦𝑦 = 𝑥) ↔ ¬ 𝑦𝑥))
1918con2bid 358 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦𝑥 ↔ ¬ (𝑥𝑦𝑦 = 𝑥)))
207, 13, 193jca 1125 . 2 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥𝑦𝑦𝑥)) ∧ (𝑦𝑥 ↔ ¬ (𝑥𝑦𝑦 = 𝑥))))
2120rgen2 3198 1 𝑥 ∈ On ∀𝑦 ∈ On ((𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥𝑦𝑦𝑥)) ∧ (𝑦𝑥 ↔ ¬ (𝑥𝑦𝑦 = 𝑥)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   ∈ wcel 2115  ∀wral 3133   ⊆ wss 3919  Oncon0 6178 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181  df-on 6182 This theorem is referenced by: (None)
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